We start with a basic Bayesian Stackelberg game with one leader type and multiple follower types. Let \(\Theta :=\lbrace \theta _1,\theta _2,\ldots , \theta _K\rbrace \)  be the set of all the follower types, and its cardinality is \(|\Theta |=K\in \mathbb {N}\) . The type \(\theta \)  follower occurs with a probability \(\mu ^*(\theta )\) , where \(\mu ^*\)  is a probability measure over \(\Theta \) . We use \([M], M\in \mathbb {N}\)  and \([N], N\in \mathbb {N}\)  to denote the leader's and follower's pure strategy set, respectively. The payoff of the leader is determined by the utility function \(L:[M]\times [N]\rightarrow \mathbb {R}\) , and similarly, the utility function of type \(\theta \)  follower is \(F^\theta :[M]\times [N]\rightarrow \mathbb {R}\) . We assume that both players' utility functions are bounded. In the basic model of Bayesian Stackelberg game, the leader would first commit to a mixed strategy \(\mathbf {x}\in \Delta _M\) . It should be noted that the mixed strategy is not mere randomization over pure strategies. Instead, it leads to a stronger commitment advantage [1]} and has attracted significant attention due to its direct applications in real security domains [2]}. After observing the leader's mixed strategy, the follower would come up with the best response, maximizing his expected utility under the mixed strategy. Without loss of generality, we assume the follower's best response is a pure strategy with ties broken in favor of the leader. Finally, the leader plays a pure strategy sampled from the mixed strategy, while the follower plays his best response, and their payoffs are determined by their utility functions. We note that the leader's realized action is effectively hidden from the follower since the follower's best response only depends on the mixed strategy.
